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Article Dans Une Revue Manuscripta mathematica Année : 2017

Regularity and rigidity theorems for a class of anisotropic nonlocal operators

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Résumé

We consider here operators which are sum of (possibly) fractional derivatives, with (possibly different) order. The main constructive assumption is that the operator is of order 2 in one variable. By constructing an explicit barrier, we prove a Lipschitz estimate which controls the oscillation of the solutions in such direction with respect to the oscillation of the nonlinearity in the same direction. As a consequence, we obtain a rigidity result that, roughly speaking, states that if the nonlinearity is independent of a coordinate direction, then so is any global solution (provided that the solution does not grow too much at infinity). A Liouville type result then follows as a byproduct.

Dates et versions

hal-03621418 , version 1 (28-03-2022)

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Alberto Farina, Enrico Valdinoci. Regularity and rigidity theorems for a class of anisotropic nonlocal operators. Manuscripta mathematica, 2017, 153 (1-2), pp.53-70. ⟨10.1007/s00229-016-0875-6⟩. ⟨hal-03621418⟩
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